How is absorption integrated in the simple Drude model?

Question

What is the physical process of absorption in the Drude model?

As far as I understood in the Drude model you only consider the electrons as classical particles and frozen ions. No other particles such as photons are present in this model. In addition, you can have external forces.

Still analytically you can get an expression for absorption in the Drude model - because absorption is usually proportional to the imaginary part of the conductivity or/and the dielectric function.

But, since you only consider electrons and ions, but no photons, I don't see where we should get absorption. Can you help me out?

Edit: I mean absorption of photons in the Drude model. Can absorption of photons be included in the Drude model?

Answer 1

The absorption coefficient and the dielectric function are valid descriptions for "macroscopic" electromagnetic fields, so you would not use these quantities to look at the absorption of single photons specifically.

For a perfect metal (infinite conduction/no scattering), the dielectric function of a metal has infinite imaginary part. Which will make the absorption coefficient infinite as well. Meaning that the electric field (from the photons) will not penetrate into the metal at all.

Answer 2

In Drude model the electron is accelerated by the electric field according to the Newton's second law $$ m\dot{v}=-eE $$ until it scatters and loses all its energy. Taking the time between collisions to be $\tau$, one can calculate the energy transferred to the lattice, as shown in this answer.

One could redo the same calculation with the ac electric field, $$ m\dot{x}=-eE(t)=-eE\cos(\omega t) $$ and obtain the energy damped into the lattice, which will now depend on frequency. This energy is the energy absorbed (one would usually average it over the oscillations period.)

Remark

• I consider simplified version of Drude model, where (a) the time between collisions is deterministic, and (b) the electrons completely lose their energy. One can use the times distributed as $w(t)=\tau^{-1}e^{-t/\tau}$, Maxwell-Boltzmann velocity distribution, and a factor characterizing the percentage of the energy transferred in collisions to make the model a bit more realistic.
• One could also do even simpler calculation by considering Newton's equation with viscous friction (instead of collisions), as Wikipedia article on Drude model suggests (although some do not call it Drude): $$m\dot{v}=-\frac{v}{\tau}-eE_0\cos(\omega t).$$